Biomedical Engineering Reference
In-Depth Information
m
according to the concise but meaningful descrip-
tion of Noldus
et al
. (1994, 1995):
∑
=
x
=
−
a
x
−
c
y
+
I
,
i
i
i
ij
j
i
j
1
y
=
f
(
x
)
i
i
•
when a neural network is used as a classifica-
tion network, system's equilibria represent
the “
prototype
”
vectors
that characterize
the different classes: the
i
th
class consists of
those vectors
x
which, as an initial state for
network's dynamics, generate a trajectory
converging to the
i
th
“
prototype” equilibrium
state
.
(2)
The first equation describes the state evolu-
tion—the state
x
of the neuron at time
t
is given
by its potential or short-term memory activity at
that moment. It depends on its inputs (the second
and third terms of the first equation) and on the
passive decay of the activity at rate
−
(first term).
The input of a neuron has two sources: external
input
I
from sources outside the net
wor
k and
the weighted sum of the outputs
i
•
when the network is used as an
optimizer
,
the
equilibria represent
optima
.
j
1=
of an
arbitrary set of neurons within the network (the
pre-synaptic neurons). The weights are some real
numbers
c
ij
, which describe the strength and type
of the synapse from the pre-synaptic neurons
j
to
the post-synaptic neuron
i
; positive weights
c
ij
>
0 model excitatory synapses, whereas nega-
tive weights
c
ij
<
0 model inhibitory synapses.
The second equation gives the output of the
neuron
i
as a function of its state. Generally, the
input/output (transfer) characteristic of the neuron
)
y
,
m
The mathematical model of a RNN with
m
neurons consists of
m
-coupled first-order ordinary
differential equations, which describe the time-
evolution of the state of the dynamical system. The
state
x
of the entire system is given by the states
x
i
,
i =
1, ...,
m
of each neuron in the network. On
other words, the state of the RNN is described by
m
state variables;
T
[
1
=
is the state vector
of RNN, where
T
denote the transpose. Using
the notation
x
for the derivative of
x
with respect
to the time variable
t
, the time evolution of the
dynamical system is described by the first-order
vector differential equation
x
x
x
]
m
(
f
is described by sigmoidal functions of the
bipolar type with the shape shown in Figure 1.
These are bounded monotonically non-decreasing
functions that provide a graded nonlinear response
within the range
x
=
h
(
x
)
,
x
∈
m
(1)
[−
. Sigmoidal functions are
globally Lipschitz, what means they satisfy in-
equalities as
1
where
T
)(
1
=
is a nonlinear vector
function which satisfies the sufficient conditions
for the existence and uniqueness of the solution
)
h
x
[
h
(
x
)
h
(
x
)]
m
f
(
)
−
f
(
)
1
2
0
≤
≤
L
,
∀
≠
tx
, for a given initial state
x
0
at a given initial
time
t
0
. In order to describe the functions
h
i
(
x
),
i=
1, ...,
m
we briefly discuss now the mathematical
model of the artificial neuron, without insisting
on the biological aspects; these may be found in
several references (see, for instance Cohen &
Grossberg, 1983; Bose & Liang, 1996; Vogels
et
al
., 2005).
Consider the standard equations for the Hop-
field neuron
(
,
x
0
t
,
1
2
−
0
1
2
(3)
Let us remark that
f
(0) = 0, what means that
sigmoidal functions also satisfy inequalities as
f
(
)
0
≤
≤
L
.
(4)
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